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Physical Review Letters

Reduced Equations of Motion for Molecular Multiphoton Processes

Shaul Mukamel

Published Jan 15, 1979• 25 citations
Publisher

Abstract

I derive reduced equations of motion for molecular multiphoton processes by constructing a set of relevant operators and adopting the Mori projection-operator technique. The intramolecular dephasing processes which play a dominant role in the quasicontinuum are properly incorporated by utilizing the representation of exact molecular states. The resulting equations include dephasing operators expressed in terms of intramolecular dipole correlation functions, and interpolate between the coherent and incoherent driving limits.

DOI
10.1103/physrevlett.42.168

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•Sep 10, 1987•0 citations
I derive reduced equations of motion for molecular multiphoton processes by constructing a set of relevant operators and adopting the Mori projection-operator technique. The intramolecular dephasing processes which play a dominant role in the quasicontinuum are properly incorporated by utilizing the representation of exact molecular states. The resulting equations include dephasing operators expressed in terms of intramolecular dipole correlation functions, and interpolate between the coherent and incoherent driving limits.
Reduced equations of motion (REM) are derived for molecular multiphoton processes by constructing a small set of relevant operators corresponding to populations and coherences of the molecule ’’dressed’’ by the radiation field and adopting the Mori projection operator technique. The resulting REM contain time-dependent dephasing operators which are intimately related to our reduction scheme and whose definition and evaluation do not rely on any perturbative expansion in intramolecular interactions. These operators are expressed in terms of intramolecular dipole correlation functions which are the key quantities in the present formulation and are then evaluated using a simple model. In the limit where our physical system may be separated into a ’’system’’ and a ’’bath’’ with a weak interaction, we recover the ordinary expressions of line broadening. The present formulation provides us with a hiearchy of REM which interpolate smoothly all the way from the coherent to the incoherent driving limits and are in good quantitative agreement with the existing experimental data.
Starting with the complete molecular Liouville equation and making use of the Mori–Zwanzig projection operator formalism we derive generalized rate equations which describe the collisionless dynamics of level populations in the quasicontinuum of molecular multiphoton processes. We define precisely the complete molecular information required for the description of these processes which is expressed in the form of a hierarchy of intramolecular dipole correlation functions. We then discuss the general conditions under which the equations attain the form of simple rate equations and show how the complete molecular information reduces considerably in this case to three relevant quantities per transition: an integrated Rabi frequency, a dephasing time and ratios of statistical weights of the levels.
In this paper we present a quantum mechanical theory of multiphoton photodissociation of large, collision-free, molecules, which rests on the notion that the radiative coupling terms between adjacent sets of congested bound molecular states in the quasicontinuum exhibit a wide variation both in terms of magnitude and of sign. Invoking the rotating-wave approximation, neglecting spontaneous infrared decay, and assuming that near-resonant radiative coupling prevails, the equations of motion were solved within the framework of the random radiative coupling model for the radiative interactions in the quasicontinuum. In the low energy range (range I) the equations of motion for the amplitudes are determined by the effective Hamiltonian formalism, while in the quasicontinuum (range II) the populations are governed by kinetic equations for sequential reversible decay. All the features of coherent excitation are preserved for range I, while in range II intramolecular erosion of phase coherence effects prevails. This model provides a set of reasonable predictions.
Multiphoton excitation and dissociation of large molecules under collisionless conditions is discussed in terms of an intercontinuum random coupling model. The mathematical approach described in a previous paper is used to obtain the general solution for a system of consecutively coupled discrete states, quasicontinuous manifolds, and continuous (dissociative) manifolds of molecular levels (eigenstates of the total molecular Hamiltonian), where the radiative coupling matrix elements are assumed to be given as a linear combination of smoothly varying and randomly varying (over level indices in the molecular manifolds) components. In the range of discrete molecular levels the time evolution is coherent and described in terms of the optical Bloch equation. In the quasicontinuous and continuous ranges the time evolution may be described in terms of Markoffian kinetic equations for the number of photons absorbed by the molecule, provided that the intramolecular vibrational relaxation widths associated with the optically active molecular modes is much larger than the Rabi frequency associated with the excitation of these modes. The kinetic evolution itself consists of direct multiphoton excitation processes (simultaneous transitions from the upper discrete levels to all higher energy molecular manifolds) resulting from the smooth component in the radiative coupling, and a consecutive excitation process described by the Pauli master equation with rates given by the golden rule expression. The interaction which enters into the golden rule expression is the variance in the radiative coupling. The direct excitation component contributes a negligible part of the overall excitation even if the random and smooth radiative coupling components are comparable. The resulting incoherent time evolution of the multiphoton excitation process is consistent with available experimental results. Coherent effects in the time evolution are expected for higher radiation field intensities, where the Rabi frequency becomes comparable to the intramolecular vibrational relaxation rate.
Reduced equations of motion (REM) are derived for the reduced density matrix of some of the degrees of freedom taking part in a scattering process. The derivation is made making use of the Zwanzig–Mori projection operator technique and utilizing a partial time ordering prescription (POP) which results in simple REM which contain no convolution in time. The present formulation is convenient for the direct calculation of reduced information on scattering events since it avoids the necessity of calculating the complete S matrix. The entire dynamical information relevant for the present description is expressed in terms of a hierarchy of correlation functions which in turn depend on the potential surface only in the region of interest. We further develop an exponential approximation for the reduced tetradic scattering S matrix in Liouville space, which forms a convenient framework for systematic approximations.
Description of detailed temporal excitation dyanmics for coherent excitation, such as is produced by idealized laser radiation, contrasts with evaluation of rate coefficients by means of generalized Golden Rule procedures; it requires an appropriate time-dependent Schroedinger equation. When the atom undergoing excitation is also affected by incoherent processes, such as collisions, this equation no longer suffices. The Heisenberg equations, or equivalent density-matrix equations, permit treatment in which coherence and incoherence play comparable roles in the excitation dynamics. Unlike rate equations, such equations must incorporate complexities that originate in the orientation degeneracy expressed by magnetic quantum numbers. In simple cases of coherent excitation, both for single-photon and multiphoton excitation, the sublevels merely require an average of 2J+1 independent Schroedinger equations. Relaxation couples the independent equations. It has been known for some time that appropriate state-multipole operators can simplify the description of many phenomena connected with optical pumping. This memo discusses application of these multipole operators to the description of Raman (or more general multiphoton) coherent excitation. In some simple limiting cases the equations simplify, but in general one has a hierarchy of coupled multipole polarizations and coherences in place of the populations and coherences that occur as variables in nondegenerate systems. 28 refs., 4 figs.